∃y q(y)

Definition

The expression $$∃y q(y)$$ is a formula in predicate logic that states 'there exists a value of y such that the property q holds for y.' This term is critical in understanding how quantifiers work, particularly in distinguishing between free and bound variables. In this context, the variable y is bound by the existential quantifier $$∃$$, which indicates that there is at least one instance that satisfies the property q.

5 Must Know Facts For Your Next Test

1. In the expression $$∃y q(y)$$, the variable y is considered a bound variable, meaning it is only relevant within the scope defined by the existential quantifier.
2. The scope of the quantifier $$∃y$$ includes everything that follows it until the end of the formula or until another quantifier overrides it.
3. This expression asserts that at least one instance exists in the domain where the property q holds true for y, making it an important part of existential reasoning.
4. If you were to replace y with a specific name or term, like 'Alice', it would no longer be an existential statement and would instead assert something specific about 'Alice' rather than an existence claim.
5. Understanding $$∃y q(y)$$ is foundational for more complex logical statements and proofs, as it lays the groundwork for discussions about existence and uniqueness in logic.

Review Questions

• How does the concept of bound variables relate to the expression $$∃y q(y)$$?
  • In the expression $$∃y q(y)$$, y is a bound variable, which means its value is defined by the existential quantifier $$∃$$. The quantifier indicates that there is at least one instance where the property q holds true for some value of y. Understanding this relationship helps clarify how different variables can behave within logical statements, particularly regarding their relevance and application within specific contexts.
• What are the implications of changing a bound variable in $$∃y q(y)$$ to a free variable?
  • Changing a bound variable in $$∃y q(y)$$ to a free variable alters the nature of the statement significantly. A free variable lacks a quantifier's binding influence, meaning it can take on any value independently. This shift transforms the expression from making an assertion about existence (that some y satisfies q) to simply describing a property without asserting that any specific instance satisfies it. Such distinctions are crucial when constructing logical arguments or proofs.
• Evaluate how understanding $$∃y q(y)$$ aids in constructing more complex logical expressions involving multiple quantifiers.
  • Understanding $$∃y q(y)$$ lays a foundational understanding for working with multiple quantifiers in logic, such as mixed expressions involving both existential $$∃$$ and universal $$∀$$ quantifiers. This knowledge enables you to assess how variables interact within complex formulas, helping you to determine scopes and relationships accurately. For instance, when analyzing statements like $$∀x ∃y p(x,y)$$, recognizing which variables are bound and how their scopes overlap becomes crucial for deciphering the overall meaning of these logical structures.

Related terms

Quantifier: A symbol used in logic to specify the quantity of specimens in the domain of discourse that satisfy an open formula. The two main types are existential $$∃$$ and universal $$∀$$.

Bound Variable: A variable that is quantified within a logical expression, meaning its value is determined by the quantifier preceding it, such as in $$∃y$$ or $$∀x$$.

Free Variable: A variable that is not bound by a quantifier in a logical expression, which means its value can vary independently and does not have a specific assignment.